This property creates a scene of conflict around you, and so is one of the most useful but dangerous features in the game.
Conflict has several effects. Peaceful or tame monsters will start attacking you or other monsters near them, though for all other intents and purposes they are still peaceful/tame, e.g. attacking them still carries consequences. This makes conflict dangerous near strong pets, shopkeepers, or other peacefuls. However, it will also make nearby monsters attack each other instead of you, which is incredibly helpful, especially in high-density areas like the Astral Plane and Moloch's Sanctum.
Specifically, monsters are affected if they are in your line of sight and at most 8 squares away. Blinded monsters and the Wizard of Yendor are impervious to conflict. Monsters with a sufficiently high combined magic resistance and level can resist this.
The exact formula for a monster resisting conflict is
rn2(100 + alev - dlev) < mtmp->data->mr, where
alev is 5 for a ring (of conflict) and 10 for a weapon (like the Sceptre of Might),
dlev is the current monster level capped at 50 and
mtmp->data->mr is the monsters magic resistance. For peaceful monsters,
alev is always 5. For example, a level 28 master mind flayer pet will always resist conflict from the Sceptre of Might, as rn2(77) will always be smaller than the flayers MR of 90.
Additionally, if you are generating conflict when you arrive on the Astral Plane, instead of getting a guardian angel you will be attacked by some hostile angels. If you aren't wearing the ring when you arrive, but later put it on and your guardian angel notices, it will also turn hostile. Conflict also has another effect: Monsters will move around randomly if they engulf you. This can be very helpful, if used properly, to a pacifist.
If you are not wearing the ring, conflict can be toggled by invoking the Sceptre of Might. This has exactly the same effects as the ring, including attack level in resistance rolls, but will spare you from hunger.
Generating conflict is a very effective way of clearing out dense groups of enemies, since monsters will fight each other in addition to you. A common strategy is to burn Elbereth on the ground, and sit back and watch as the monsters battle it out while you take minimal damage. All monsters are prone to conflict (some may resist as explained in the formula above,) including normally peaceful monsters and your pet, however, you can still incur the penalty of murder on the peaceful humans/kops that attack you! Elbereth will continue to work against those who respect it as long as the engraving remains intact. Popular places for conflict are crowded levels such as the Big Room and certain quest branches. The elemental planes are also a good candidate for conflict, though do note that on the Astral Plane, if you are generating conflict, you will be sent hostile angels instead of a tame one, and generating conflict at any time will instantly replace the tame one with hostile ones.
One important thing to consider is that no experience is gained by the player from monsters that kill each other, but monsters will gain experience for themselves. Use caution when generating conflict in the presence of monsters that tend to grow up into more powerful forms, since they can rack up experience from killing off the lesser monsters. If you are sufficiently high level, this is usually not much of a problem. Also take care when generating conflict with priests or shopkeepers present. As mentioned above, they can and will attack the player while under conflict and you will still get the usual murder penalties if you kill them.
Be careful when skipping through all of the battle messages whilst generating conflict, as it is easy to skip over important messages pertaining to, for example, a cockatrice hissing at you and beginning to stone you. Always pay attention to the monsters around you and any status effects that you might have gained during the flood of "The <monsters> hits <monster>!" prompts.
Rings of conflict are the usual way to use conflict, by wearing and removing the ring as necessary. The safest method of conflict, from a nutrition standpoint, is the Sceptre of Might: You won't get hungry, but you also only get to turn it on every once in a while. Thus, the Sceptre is best used for emergency crowd-clearing; e.g. if a monster summons nasties.
Eating a ring to get intrinsic conflict is usually a bad idea as it will burn through nutrition very quickly, and you will have no way of shutting it off. If you ate the ring by accident, your only option is to try removing intrinsics at random, by eating the rare disenchanters or letting a gremlin attack you at night.